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NATIONAL BUREAU OF STANDARDS REPORT
10343
PROJECTS and PUBLICATIONS
of the
APPLIED MATHEMATICS DIVISION
A SemiAnnual Report
July through December 1969
U.S. DEPARTMENT OF COMMERCE
NATIONAL BUREAU OF STANDARDS
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BOREAii 8F STfi^BARDS REPOIIT
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IMPORTANT NOTICE
NATIONAL BUREAU OFfor use within the Governmei
and review. For this reason
whole or in part, is net eu
Bureau of Standards, vVashi
the Report ha: been specific
Approved for public release by theDirector of the National Institute ofStandards and Technology (NIST)
on October 9, 2015.
)gress accour.ting documents intended
it is subjected to additional evaluation
tore listing of this Report, (jither in
the Office of the Director, Nation.il
or, by the Government agency for which
tal copies for its ov/ii use.
U.S, DEPARTMENT OF COMMERCE
NATiOKAl. OUREAll OF STANDARDS
A.PPLIED MATHEMATICS DIVISION FUNCTIONS
Conduct research and provide consulting services to the Bureau and other Federal agenciesJn various fields of mathematics important to phy.sical an.d engineering sciences, automaticdata processing, and operations research with emphasis on statistical, numerical andcomhinatorial analysis, and mathematical physics. Develop tools for mathematical work suchas mathematical tables, handbooks, manuai.s, mathematical models and computational methods,and advise on their use. Provide training in disciplines related to these functions.
NUMI’.RICAL ANALYSIS SECTION: The advancement of computation and the theoryof munerical analysis, particularly in the development of computing algi,rithmapproximations to functions, end methods to facilitate the use of high sp'^delectronic computers by subject matter specialists. Design of mathematicaltables; exploratory calculations on automatic machines. Consulting servicesand training, and preparation of manuals in these fields. Research inunderlying branches of pure and applied mathematics, such as matjix algebra,combinatorial analysis, and number theory.
OPERATIONS RESEARCH SECTION; Development and application of mathematical andcomputational techniques for the analysis, improvement or optimization >fcomplex systems or activitypatterns. This includes (l) research in specificrelevant areas of mathematics, such as linear programming, the theory of linegraphs, and the theory of strategic contests, (2) investigations in the artof constructing useful mathematical models of complex systems, and ofobtaining information about the system by applying analytic or simulationmethods, and ( 3 ) application of these techniques to selected problems,of general methodological significance, arising in the work of the Bureauor of other Government agencies lacking specialized pei'sonncl in this field.
STATISTICAL EJCINEERIMG SECTION: Consulting services in the applicationof mathematical statistics to physical science expei'iments and engineeringtests, particularly in the design of experiments and in the analysis andinterpretation of data. Research on pertinent topics in probabilityand mathematical statistics. Preparation of reports, manuals, tables,studies of computational methods and other aids to the application efmodern statistical methods .
SYSTEIIS DYNAMICS SECTION; Research and consulting in applied mathematicsbasic to physics and engineering, with emphasis on analysis of the dynamicbehavior of complex physical systems. This involves, primarily, thedevelop.ment and application of techniques foi’ solving linear and nonlinear system.s of differential equations and integral equations, orcombinations of both. Of concern also is simulation of the behavior ;fphysical systems by means of electronic computers using approximationtechniques and semianalytic methods. Attention is given to problems inplasma dynamics and the behavior of solid matter and multi compone.'nt liquidsyste.ms, with emphasis on developing matliematical methods of wide rangeof applicability beyond the scope of the immediate problem. Invcstig.atl nisare carried out on the special functions encountered in the analysis and
algpritliras for their evaluation are prepared.
II
Contents
Status of Pi’ojects° as of Becembei' 31,1969 ^
1. Numerical analysis 1
2. Operations research ‘i
3 Probability and mathematical statistics 8
4. Statistical engineering services 10
5 . Systems dynamics
Lectures and technical meetings 1?
Publications activities 19
°0nly unclassified material is included in this report.
Ill
APPLIED MATHEMATICS DIVISION
Jjly 1, 1S69 through Dacernber 31, 1969
TECHNICAL ADVISORY PANEL
Marvin L» Minsky, Massachusetts Institute of Technology
Carl A, Bennett, Battelle Hsmorial Institute Murray S. Klamkin, Ford Scientific LaboratoryCharles R. DsPrima, California Inst, of Technology M. Ho Martin, University of MarylandH, Oo Hartley, Texas A & M University John Riordan, The Rockefeller University
John Todd, California Institute of Technology
DIVISION OFFICE
Edward W, Cannon, Ph,D,, Chief
Catherine Hartsfleld, Secretary Molly F, Hevonor, Admin. Officer, B.S."Yates S. Bladen, Admin, Officer®
Russell A. Kirsch, M.S. A.nne A. Holston, B.A. Irene A. Stegun, M.A.
Margaret L. Dean Gertrude V, Thompson Ruth Zuclcer, B.A.Ida I. Rhodes, M.A. , Consultant
NlRklERICAJ. ANALYSIS SECTION, Morris Newman, Ph,D., Chief
Doris M, Burrell, Secretary Joseph Lehner, PhtD,"*"
Karl Goldberg, Ph.D, Russell L. Merris, Ph.D.Seymour )!aber, Ph.D.
Frank W, o. Olver, D.Sc.Stephen J. Pierce, Ph.D,Justin C. Schaffert*
STATISTICAL ENGINEERING lABQPJlT'OPY, Joan R, Rosenblatt, Ph.D,, ChiefHslou K, liu, PhcD., Ase‘t Chief
Raj Cliandra Bose, Pli.D,, ConsultantShirley G, Bremer, B.A.>'''Eleanor S, BrownVeronica Connor, SecretaryJames J. Filliben, Ph.D.
David Hogben, Ph.D,
Brian L, Joiner, Ph.D.Mary G, Natrella, B,A,>!‘
Sally T. Peavy, B.A.'^^
SYSTEMS DYNAMICS SECTION, Hans J. Oser, Ph.D,, Chief
Jeffrey T. Fong, Ph.D.Regina Kopec, Secretary
S, Richard Kraft, Ph.D,Daniel W. Lozlei', M.Ai,
Malcolm S. Scott, M.S.’’'Janace , Speckman
, M.S.Ruth N. Varner, B.A.*Roy H. Wampler, M.A.
William H. Youden, Ph.D.^+*
Zol]a G, Ruthberg, M.S.’f'
Walter L. Sadowskl, Ph.D.
OPERATIONS RESEARCH SECTION, Alan J. Goldm.ati, Ph.D., Chief
Jack Edmonds, M,A,°
Judith F. Gilsinn, M.S.William C. Hall, B.A.Dolores F, Harrison, SecretaryWilliam A. Horn, Ph.D,
liSmbort S. Joel, B.A,
David Klavan, B.A.’i’
David S, Lfibovitz, B.A, “S'
Jeffery C. Lagarlns
Joel l,0vy, M.A.
Pliilip Meyers, Ph.D.t
Martin H. Pearl, Ph.D.*Patsy L, Saunders, B.S.Roger D, Traub, B.A,*
*Partt imo^^Postdoctoral Research Associate***Guest Worker
°On Leave of Absence^E.O.D, November 5, 1969^Retired October 31, 1969
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Status oi Projects
1 . NUMERICAL ANALYSIS
RESEARCH IN NUMERICAL ANALYSIS AND RELATED FIELDS
Task 20501122050110/5555
11540
Origin: NBSManager: Morris NevananFull task description: July  September 1954 issue, p. 1
Authorized 8/29/54
Status: CONTINUED. M. Nev.mian has proved that if 77 is a unit in the cyclotomic number fieldR(C) > where ^ is a primitive pth root of unity and p is a prime > 3 , then Tj is the rrh
power of another unit only if1
rj < log N/log(2 cos^ ) ,
where N is the maximum of the absolute
values of 17 and its conjugates. This result was applied to prove that a number of diophantineequations involving cyclotomic units had only trivial solutions.
M. Newman and L. Greenberg have proved that the density (in a welldefined sense) ofsolvable groups generated by elements of odd order is zero.
M. Nc^'/man has prepared several versions of his "exact solutions" program using only basicFORTRAN. These can be used immediately on any machine with a word length of 36 or more bits. Somedozen copies of this program have been distributed by request.
S. Haber completed vjriting a survey of the literature on numerical evaluation of m.ultipleintegrals
.
F.W.J. Olver has continued the study of the method of .stationary phase for the asymptoticexpansions of oscillatory iiAtegrals. An improved form of error analysis has been constructed and ibeing tested on illustrative examples.
K. Goldberg extended his results on similarity for formal power series over fields,substituting a matrix equation for the determinant equation previously found.
K. Goldberg extended his tables of the coefficients in formal power series for variousoperations including product, iteration, inversion, and the related power series L^(z) defined
by L^(f(z)) = L^(z) • f'(z) .'
S. Pierce has determined all multiplicative maps of the 2X2 matrices over a Dedekindring which do not send all singular matrices to 0 .
S, Pierce has proved that all members of the orthogonal group of a real positive multilinear form have only eigenvalues of modulus 1 and only linear elementary divisors.
1
Status of Projects
S, Pierce and K. Merris have shora that associated transformat ioiiw on higher degreesymmetry desses of tensors regarded as representations of the full linear group are alwaysreducible. Investigations for the case of degree one symmetry classes arc continuing.
S. Pierce has given a short proof using the Hilbert reciprocity law that ] is a simiof tv;o squares in any cyclotoralc field of order m when in is divisible by a prime
p s 3 or 5 (mod B)
R. Merris discovered a generalisation, T,of the trace of a matrix. If A and B
are complex mXn matrice.s, (A,B) “ T(B'''A) is a positive semidefinite hen.nltian (psdh) form .Suppose H = (H^j) is an mnXmn block psdh matrix where is niXm . Then (T(l'^^)) is [)cdh .
R. Merris and S. Pierce proved the space of mXn matrices with the generalized traceform is regular on].y if T is the trace. Research in this area is continuing.
S, Pierce and R. Merris proved that if A is a complex matrix of sufficiently largerank and if all the elementary divisors of K(A) are linear then all the elementary divisors ofA are linear. Here K(A) is an as.soclated transformation based on a group and an irreduciblecharacter of arbitrary degree.
R. Merris and U. Watkins discovered some conditions forcing isomorphism between symmetryclasses of tensors.
R. Merris discovered an inequality for generalized matrix functions and block herndtianmatrices
.
Publications :
(1) Stochastic quadrature formulas, S. Haber. To appear in Math. Comp.
(2) Sequences of numbers that are approximately completely equidistrlbuted . S. Halier. Submittedto a technical journal,
(3) On the sum £< C n > ^ and numerical integration. S. Haber and C.F, Osgood. To appear inPacific Jour, of Math.
(4) Numerical evaluation of multiple integrals. S. Haber, Submitted to a technical journal,
(5) Lectures on modular forms. J. Lehner. IiMS 61 of the NBS
.
(6) AuLomorphic Integrals vjith preassigned periods. .1, Lehne.r , J. of Re.^carch NBS, 73.B, 153161
(1969)
.
(7) On the multipliers of the Dedekind modular function. J. Lehner. J. of Research NBS, 72B , No. 4,253261 (1968) ,
(5) Subgroups oi SL(t,z). M, Newman. J. of Research N8S, 73B, 143146(1969).
(9) Some results on unitary matrix groups. M. Newmtm and M. Marcus. To appear in J. of Linear
Algebra
.
(10) Some results on roots of unity, with an application to a diophantlne problem, M, Newman.
Aequationes Hath. 2, 163166 (1969).
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(11) Isometric circles of congruence groups, M. Ncx'nnnn, Amer. ,1. Matn. 91, 648656 (1969).
(12) Principal ideals in matrix rings (v;ith S. Pierce). M. Mewman. J. of Research tlBS, 733,211213 (1969).
(13) Normal subgroups of the modular group. L. Greenberg and 11. liewman. Submitted to a technicaljournal .
(14) A diophantine equation, M. Kevtman. J. London Math. Soc. , 43, 105107(1968).
(15) A table of the first factor for prime cyclotomic fields. M. Nevroan. To appear in Math, ofComputation
.
(16) Some results on solvable groups. L. Greenberg and M. Newman. To appear in Arch.der Math.
(17) On Riemann surfaces with raaxim.al automorphism groups. J. Lehner and M. Newman. GlasgowMathematical Journal, Vol. 8, Part 2, 102112 (1967).
(18) An enumeration problem for a congruence equation. M. Ne’tman and R. Brualdi. To appear inJ. of Research NBS
,
(19) Why steepest de.scents? F.W.J, Olver, To appear in Proceedings of 1969 SIAM National Meeting,June 1012, 1969.
(20) Bounded matrix groups. M. Net'/mari and S. Pierce. To appear in Journal of Linear Algebra.
(21) Orthogonal groups of positive definite multilinear functionals. S. Pierce. To appear inPacific Jour, of Math.
(22) Multiplicative maps of matrix semigroups over Dedeklnd rings. S. Pierce. Submitted to atechnical journal.
(23) Orthogonal decompositions of tensors spaces. S. Pierce. To appear in J. of Research NBS.
(24) Trace functions I . R. Kerris. Submitted to a technical journal.
(25) Elementary divisors of higher degree associated transformations. R. Merris and S. Pierce.Submitted to a technical journal.
(26) A class of representations of the full linear group. R, Merris and S. Pierce. Submitted toa technical journal.
(27) Isomorphic symmetry classes of tensors. R. Merris and W. Watkins. Submitted to a technicaljournal .
'
(28) Partitioned herraitian matrices. R. Merris. To appear in J. of Re.searc,h NBS.
3
2. OPKRyVnOKS RESEARCH
CONSULTATION IN ILViTlEMATICAL OJ'ERATIONS RESEARCl;
Task 205122050151
Origin and Sponsor: NBS 11570 Authori:^ed 12/30/60Manager: A.J. GoldmanFull task description: OctoberDecember 1960 issue, p.3
Status: CONTINUED.
(1) Demand for miscellaneous consulting and advisory services remained heavy. Section staffprovided such services in 99 recorded instances, 96 involving assistance to NBS staff. The 99ins(;ances totalled to 695 recorded manhours. Cither agencies assisted included the Civil ServiceCoiirmission, National Science Foundation, Office of Emergency Planning, Maritime Administration,Law Enforcement Assistance Agency, and Bureau of the Budget. Requests from universities, industry,professional groups and journals v;ere also met.
(2) J. Gllsinn continued collaboration with the Information Processing Technology Divisionin a study, for the Federal Aviation Authority, to improve air traffic designation procedures. ApreviouslyX";ritten simulation program was documented and applied to test an initial scheme forassigning radar beacon codes to airplane flights; the results were analyzed and documented. (Reportedhere for convenience; supported under Project 6505453.)
(3) A.J. Goldman, W.A. Horn, J. Levy and M.H. Pearl completed a collaborative effort with theTechnical Analysis Division on another study for the FAA, this one to analy.ze a proposed nevj"capacity" concept for an airport runway and the associated finalapproach airspace. For a singlestream of customers, the concept w’as shown (using tl'.e theory of Markov renev.iai process) to be representable by a simple formula useful for planning and analysis purposes. The theoretical development and illustrative calculations were documented, and also presented orally at several briefingsfor interested FAA personnel. (Reported here for convenience; supported under Project 4314449.)
(4) W.G. Hall continued helping the Army Data Field Systems Command in planning, design andnumericalanalysis aspects of a new tactical artilleryfire control system. Efforts includedcontractor monitoring, and design and testing of a short course in basic computer programming andnumerical analysis for tho Fort Sill Artillery Training School. (Reported here for convenience;supported under Project 6505425.)
(5) Goldman, Hall, Horn, and L.S. Joel continued assistance to the Post Office Department'sBureau of Research and Engineering in technical Interfacing v;ith contractors concerning computerized
models and data analysis. P. Saunders began assistance to the Technical Anaiysl.s Division in asimulation study of Coast Guard search and rescue operations. (Reported here for convenience;
supported under Project 4314561.)
(6) A.J. Goldman began service as Associate Editor of tlie Operations Researcli Society's"Transportation Science" journal, and ViceChairman of the Society's Transportation Science Section.
Publications
(1) M. Aronoff (Dlv. 43]), A.J. Goldman al^. Analysis of a capacity concept for runway and
finalapproach path airspace. NBS Report 10 111, 11/69.
(2) R.D. Elbourn (Div. 650) and J.F. Gilsinn. Sin:ulation of Air Traffic Control radar beacon code
assignment plans (final report on Phase 1). NBS Report 10 117.
(3) A.J. Goldman and P.R. Meyers. Slmultane ous contractif ication. Journal of Research NBS,
73B (1969), pp. 301305.
(4) A.J. Goldman. Systems analysis and urban problems (Discussion). To appear in Prcc. Sjnnposium
on Systems Analysis for Social Problems, 5/69,
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W.A. Horn. Some fixed point theorems for compact maps and flows In Banach spaces,in Trans. Ainer. Hath. Soc.
To appearC3)
(6) W.A. Horn
(7). W.A. Horn
Convex homotopy. Submitted to a technical journal.
Optimal design of sorting networks. NBS Report 10 146 (1/70).
COMBINATORIAL MliTHODS
Task 205122050152
Origin and Sponsor: NBS 11540 Authorized 12/10/60Manager: A.J. GoldmanFull task description: OctoberDecember 1964 issue, p.3; AprilJur.e 1962 , p.l5
Status: CONTINUED
(1) J. Gilsinn (and C. Witzgall of Boeing Scientific Research Labs) continued experimentsand documentation on the comparison of shor testpath algorithms.
(2) A.J. Goldman continued studies relating to the optimal location of facilities in networks.He and C. Witzgall generalized the theorem that a single node, which produces half or more of thetotal flow, must itself be an optimal location.
(3) L.S. Joel initiated computer implementation of J. Edmonds' algorithm for minimumcardinalitypartition of a matroid into independent sets. The program "frame" for general matroids, and thesubroutine to make it applicable to graphical matroids, were programmed.
Publications
(1) J. Edmonds and D.R. Fulkerson (RAND). Bottleneck extrema. To appear in J. Combinatorial Theory.
(2) J. Edmonds and 0. Shisha (WrightPat terson A.F.B.). Acute bijections. To appear in J.Combinatorial Theory.
(3) A.J. Goldman. Optimal locations for centers in a network. Transportation Science 3 (1969),pp. 352360.
LINEAR AND NONLINEAR PROGRAMMING
Task 205122050153Origin and Sponsor: NBS 11540 Authorized 12/30/60
Manager: W.G. HallFull task description: OctoberDecember 1960 issue, p.3
Status: CONTINUED
i (1) W.G. Hall and P. Saunders continued work to improve the capabilities for linear programming
:calculations at NBS. Testing has revealed flaws in several proposed programs; particularly hazardous
[
are those in which an erroneous solution (or an incorrect "no solution" indicator) arc given v.Mthout
warning.
! (2) A.J. Goldman continued studying the minimax error selection of incom.pleteiy specified
? discrete probability distributions, examining the problem of disaggregating an exactly or
[
approximately "given" distribution.
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(1) A.J. Goldman. Minlinay. error selection of a univariate distribution '.»'itli pre.scrjbcdcomponentwise bounds and ranking. Journal of Research HBS, 73B (1969), pp. 225210.
(2) A.J. Goldman. Minimax adjustment of a univariate, distribution tc saci.sfy componentwise boundsand/or ranking. Journal of Research KBG, 7 3B (1969), pp. ^3]239.
(3) W.A. Horn. A theorem on convex hulls. Journal Research NBS , 73B (1969), pp . 307308.
SCHEDULING .5ND ROUTING IN INTEKURB/W TRANSPORT
Task 205122058656
Origin: Technical Analysis Division, NBS 21550Sponsor: Northeast Corridor Transportation Project, Dept, of TransportationManager: P.B. Saunders
Full task description: This is one of three projects into which our wor.k for the Northeast CorridorTransportetion project has been split. .Its objective Is to cont.inue thedevelopment, computer impl craentetlori and nppl Icution of metjiods i,o c/Dt;iri"good" operating rules for transport systems.
Status: NEW
(1) P. Saunders continued work on integrating rail scheduling a.lgorichms and demand forecastingmodels. Contractor reports on rail and airflight scheduling were reviewed.
(2) D. Klavan programmed and debugged the algorithm developed by G. Nemhauser (Johns Hopkins U.)for jointly optimal scheduling of local and express service along a line of stations. A slightly
revised version, to liandle two "locals" which compete over some Interior links of the line, wasalso developed.
Puhl
i
cation
(1) D.R. Young (now of Stanford U,). Passenger transportation scheduling. NBS Report 10 069 (6/69).
PASSENGER DEMAND MODELS FOR INTERURBAN TRANSPORT
Task 205122050657
Origin: Technical Analysis Division, NBS 21550Sponsor: Northeast Corridor Transportation Project, Dept, of TransportationManager: A.J. Goldman
Full task description: This is one of three projects into which our work for the NortheastCorridor Transportation Project has been split. Its objective is to develop,
calibrate and prototypetest improved passengerdemand forecasting models.
Status: NEW
(11 L.S. Joel began an analysis of the promise for niodel use of various additional explanatory
variables describing characteristics of the transport system. He and .1. Lagarias initiatoil an
investigation of the relationship between intercity passenger trip volumes and telephonecall
volumes
.
6
(2) A.J. Goldman and R. Ku (Division 431) examined tv;o mo’del inodif icaLion:; : one reflecting
the idea that autodriver fatigue increases more than linearly with trip time, the other revising
the modalsplit submodel to incorporate a probabilistic lower limit on acceptable conductance.
Neither change significantly Improved the fit to the present da(:a set.
(3) J. Gilsinn, A.J. Goldman and R. Traub revie\;ed contractor reports and related literature.Ku completed a summary ch.apter on the model cui rently used by the Corridor Project; Goldman (withli. Cheslow of DOT) began a more extensive documen tu t ion
.
Publication
(1)
J. Gilsinn. Roles of demand models in the Northeast Corridor Transportation Project.NBSNDCTP Working Paper No. 12, 7/69.
MISCELLANEOUS 1LATH SERVICES CONCEPaNING INTERURBAN TPvANSPORT
Task 205122058458
Origin: Technical Analysis Division, NBS 21550Sponsor: Northeast Corridor Project, Department of TransportationManager: A. .J. Goldman
Full task description: This is one of three projects into which our work for the NortheastCorridor Transportation Project lias been split. Its objective is to providemiscellaneous mathematical services and exploratory research for theNortheast Corridor Transportation Project (NECTP)
.
Status: NEW
(1) J. Levy and M.H. Pearl continued developing methods for evaluating feedback vs. nonfeedback flow regulation at a merge point in a transport network. A comparison of the performanceof nonfeedback, feedback, and "prophesying" control v:as begun.
(2) VI. A. Korn completed studying an approach to the design of transport networks when
minimizing the number of access points is of prime concern. He also assisted in review of contractor
documents
.
(3) A.J. Goldman assisted in editing the first draft of the NECTP 12/69 Report.
Publications
(1) J. Levy and M.K. Pearl. Feedback regulation of traffic at an_ intersection of two streams.NBSNECTP Working Paper No. 11 (9/69).
(2) M. Krakowski (Tulane U.). Adaptive svritching and routing in transport networks. NBSNECTP
Working Paper No. 13 (10/69)'.
(3) W.A. Horn. Optimal Networks Joining N Points in the Plane. Strassenbau und
Strassenverkehrstechnik, ^ (i969), (Beitrage zur Theorie des Verkehrsf losses) , pp . 161166.
3. rnOP.ABILITY AKD I'OTfDiJ.lATICAI, STATISTICS
RESEAIiCH IK PROE/>EILITi' .ARD MATHEMATICAL STATISTICS
Tasic 201j03]220'^0131/63I2>91150
Origin: NBS Authorized 10/]/62Manager; Joan Raup RoaenLla.ttPull tasK description: July  December I962
Status; COlWIl'IUED. Brian L. Joiner and Eleanor S. Brown, in coDlaboration with Bert Levy (HarryDiamond labs.) and N. F. Laubscher (National Research Inst, for Math. Sciences, South A.frica),completed the preparation of a co;nhined author and permuted title index for seven statistics journals,Including articles published since the most recent cumulative subject index to each of the journals.The index V'as premred using a computer program vn’itten by the late V. W. Youden (NBS Center forComputer Sciences and Technology).
James J. Filliben is completing a study of the nature of the percent point function (ppf)the inversecumulative distribution function. It is sliown that a nuniber of statistical procedures liave a simplerand more natural expression in terras of the ppf rather than in terms of the cumulative distributionfunction. Various simplifications in probability theory and order statistic theory are presented.The use of the ppf in random number generation and probability plots is discussed. An alternativeset of measures of distributional characteristics based on the ppf, rather than on the mom.ents, isIntroduced. The heliavior of the ppf under various tj'pes of transformations is considered. The useof the ppf in generating distributions with pre .'specified tail lengtli is demonstrated.
Publi cat ions:
(1) The median significance level and other small sample measures of test efficacy. B. L. Joiner.J. Amer. Statist. Assoc., ^ (1969), 9T1985
(2) Studentt deviate corresponding to a given normal deviate. B. L. JoinerJ. Res. MBS  C. Engineering and Instrumentation, 75C (1969)? 1516.
(3) Some properties of the range in samples from Tukey's symmetric lambda distributions.Brian L. Joiner and Joan R. Rosenblatt. Submitted to a technical journal.
(^) An author and permuted title index to selected statistical journals. Brian I,. Joiner,N. F. Laubscher (Nat'l. Res. Inst, for Math. Sciences, S. Africa), Eleanor S. Brown, andBert Levy (Harry Diamond Labs., IJ. S. Amy). To appear as NBS Special Pub, 321.
(5) Approximating discrete probability distributions. H. H. Ku and S. Kullback. (G.H.'j.).IEEE Transactions on Information Theory, IT1 5 (19d9)>
(6) Analysis of multi dimensional contingency tables. H. H. Ku, R, N. Varner and S. Kullback(George Washington Unlv.). To appear in Free. l4tli Conf. on Design of Experiments in ArmyResearch Development and Testing. ,
(7) An application of minimum discrimination information estimatiojn to a problem of Grizzle andBei'Kson on the test of "no interaction" lOTothesis. H. H. Ku and S. Kullback (G.W.U.).Submitted to a technical journal.
(8) Symmetry and marginal homogeneity of an r x r contingency' table. C. T. Ireland, S. Kullbaci,(G.V’.LI.), aivi H. H. Ku. To appear in J, Araer. Statist. Assoc,
(9) An evaluation of linear least square.? computer programs, Roy H. Wampler.
J. Res. KBSB. Math. Sciences, 73B (1969), 5990*
(10) An evaluation of linear least squares computer programs: A sumuary report. Roy il. Wampler.
To appear in Proc. l4th Conf. on Design cf Experiments in Army Researcli Development and Testing.
(11) A report on tlie accuracy of some widely used least squares computer programs. Roy H. V/ampler.
To appear in J. Amar. Stati.st. Assoc.
6
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DSrvTELOPMKWT OF "OfflJlTAB"
Task 2050312205013111550
Origin and Sponsor: KBS Authorized II/1/68Managers: David Hogben, Sally T. PcavyPull task description: July  December I968
Status; CONTIIiUED. David Hogben, Sa]jy T. Peavy, Ruth N, Varner and Shirley G. Bremer continuedtrork on the development of the OMNITAB system. M. Stuart Scott developed a comiriand forcorrelation analysis detailed autonatic printing. Fight new versions have been implementedsuccessively. Details of the changes made appeared in the t'nree Newsletters issued. A set oftvrelve instructions for perforniing operations on magnetic tape vrare added to the system.R. C. McClennon (Office of Standard Reference Data) contributed two conunands for array operations.Other commands were added for printing data, and Gauss quadrature.
Many improvements were made in the printing and curve fitting commands. Results are now' normallyprinted according to the "readable" format. The automatic printing for the least squares instructions has many valuabi.e additions, in particular, one page containing four different plotsof standardized residuals.
An important change W'as made in the word length of alpha/numeric data so that ONiNITAB is now' asmachine independent as the state of the art will pennit.
Publications:
(1) The use of OMNITAB for statistical analysis of tabular data. Joan R. Rosenblatt,Brian L. Joiner, and David Hogben. To appear in Census Tract Papers: Papers Presentedat the Conference on SmallArea Statistics, Aunerican Statistical A.ssociation, New York,August 21, 1969
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COLLABOMTIOM ON GTATISTIOAJ. AGPECL'S Or'
NBS RESEARCH AMD TESTE'IG
Task 13911612050950/5.1J.99500
Origin; NBS Authorised 7/1/5OMauagei’S; H. il. ?ai. J. R, RosenblattFull task description: July  September 1950 issue, p. 60
Status: COMTINIIED. This is a continuing project involving cooperaticn VTitli other Bureau scientistson the statistical aspects of their investigations. These services vary from short (onehour)sessions to extended collaborations involving several ina.nmonthsj and are concerned prlmariDy withstetistical design of experiments, analysis and interpretation of data, and the use of cornpiuters instatistical analysis of data. Typical examples of the services performed are the following.
B;riaa L. Joiner is on detail to the Boulder Jjaboratories for one year, at the request of the NBSOffice of Measurement Services. His primary mission is to heip them establish cooperative calibrationprogratas and to provide better inforination to support the uncertainty statements on the reports ofcalibration. So far effort lias concentrated on the analysis of existing data and the planning ofexperiments to obtain f'urtlier data on oo8,xlal and microvfave bolometers.
Brian I,. Joiner presented a talk on "Error Analysis" ac a Iteser Povrer and Energy Measurement Seminarsponsored by the Quantum Electronics Section of the Radio Standards Physics Division.
Brian L. Joiner assisted Johi: Dean and Douglas Mann (Cryogenic Metrology Section) in the planning andanalysis of experiments to evaluate cryogenic flow meters.
Th.r Statistical Engineering Laboratory continues to distribute within the Bureau occasional notes onuseful subroutines and documentation aids, for computer users who are not primarily computer•pecialists. Three recent notes, for example, describe and give details for using plotting subrcjutines and programs applicable with equijjmerit available at the Bureau, Sally T. Peavy andPvUth N. Varner are the chief contributors to the series, but notes have also been ’.nritten by other(ijembers of the Statistical Engineering Laboratory and by persons in other Btireau divisions.
Roy K. V/ampler and Joan R. Rosenblatt are collaborating with G. L. Hov.'ett (NBS Office of Color: ruetry)on the analysis of the latest in a series of cooperative paired comparison experiments conducted bythe Committee oi: Uniform Color Scales of the Optical Society of America.
Joan R. Rosenblatt collaborated v.'ith M. A. Bond, Jr. (NBS Library) to develop and execute a simplesam'ole design to estimate the distribution by publ.icat 1 on date of the library's books (not includingjournals)
.
G. llatrella presented three lectures on "Statistics of Measurement" for senior tecimicians fromihe State Weights and Measures Departments of Havra.ii, Ca li.foriiia, and TiUnois, Staff meiiibers of the;iBS Office of 'Weights and Measures also attended tlie lectures.
lar.ies J. Fllliben collaborated with George Hicho (312.0j)and Harvey Yal.owltz (312,03) in the
pneT'aration and testing design of specimens for Xray diffraction standards  about 1000 pellets
compected from 5 vreight percent 310 stainless steel powder and 95 weight percent 430 stainless steel
pevder
.
pjriications;
(l) Iriteriaboiatory comparison of the potential heat test methed, D. Gross (Fire Research Section)
and M. G. Natrella. Submitted to a technical journal.
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BTATISTJCAL SERVICES
Task 20503^02050132/583^611550
Origin and Sponsors; Various Agencies Authorized 3/31/58Manager: J. R. RosenblattFull task description: January  March 1958 issue, p. h’y
Status: CONTINUED. This is a continuing project which involves providing, upon request, statistica.lservices to other governinental agencies, universities, industrial organizations, and otlier nongovernmental agencies. Approximately jO such requests are lie.ndled per month langing frcm shortconferences to coi.laboration involving several da,ys vorK,
Janace A. Speckraan used OMIITAB to perform l44 least soiiares straight line fits to data on theFederal Court of A.ppeals workload, in response to a request from James A. McCafferty of the Divisionof Procedural Studies and Statistics, Administrative Office of the United States Courts.Projections for 1970 to 1975 were computed and plotted for all 11 circuits combined and for eachcircuit separately for 6 variables  (total nuniber of appeals, numbei* of prisoner petitions, etc.)using data from i960 to 1969. Projections vrere also made using only the data from I965 to I969 tosee if the trends have changed.
The Statistical Engineering laboratory vras asked by the NBS Center for Computer Sciences andTechnology to participate in one of a series of orientation meetings for a group of foreign trainees(sponsored by the Agency for International Development) from the Census Bureau's Office of International Statistical Programs. The group included several computer prograinmers with i^articulai’interest in statistical computations.
In response to requests, twentytwo copies of the OMNITAB program were distributed.
SYS'l'i'MS UYNAMLCS
rese«jk;h lij’ sYS'i'W'is /rayuiicf.!
A!MD RliLATED PMEI/iJG
Taek ?05)i01220i)0l)^l/5557
Oi’igin: IJBS Authorli;cd 9/l/;)4
Manager; H. J. OserFull taslc description: Julygeptemler 195^i issue, p. 2"{
In consultation with Hr. J. A. Siiaaons of the Metallurgy i)iviBion (31205) and Jlr'. E. A. K'’>i2cley ofthe Mechanics Di\^slon (21305), J T. Fong proved tliai the problem of tlie finite defonaation of amemorydependent .material continuum of the BernsteinKenrsleyZapas tj^ic: (Trans. Soc. Rheology, 7,391“^10 ( 1963 ) is equ.ivelent to a variational problem for a rlieonomic system, i.c;. the local Lagrangiondepends on the time t explicitly. A manuscript entitled "On the existence of an action principle fera dissipative continuum" is hoing prepared for review and di.scussJon.
In collaboration with Mr. L. J. ZapjiS of the Mechanics ilivision ( 213 . 05 ), J. T. Fong showed thot theclassical theory of plasticity in which the time t is not a physical variable but a maihcmiatlcalparameter, can be modeled a.fter the BernsteinKearsl cyZapas ' theory cf a memory dependent materialprovided tlie elastic potential of the EKZ theory Ic a homogeneous fujict ion of the vculable t ' ofdegree 1, where t' is the difference between the current time and some time in the past. Work isC'.ontinviing to derive some basic features of a special pl.a,£;ticity theory from a cla.es of the memorydependent elastic potentials of the bKZ. theory.
Progress is al.so being made in a longtcrTu project to find Ih.e necessary end sufficient conditions I'orthe equivalence of the geometric theory of an "imperfect" material continuum due t.o Kondo, Eilby,Kroner, etc. and the historydependent kineii'.atic theory of a "sljnple" material due to Truesdell, Hell,Kang, etc. To discuss some of his f.lndings, J. T. Fong visited Professor Y. C. Wong of the Depart.me.ntof Mathematics, Uni.versixy of Horxg Kong, as a guest worker with the title of Vlslcing Research Fellowfrom July 5 to July 12, I.969 during which J. T, Fong also delivered the following two talhs
;
"Microscopic Theories of Engineering Material.s"  University of Hong Kong;Department of Civil Engineering Seminar, July E, I969 ;
"Applications of Differential Geometry in Continuum Physics"  Universityof Kong Kong: Dept, of Mathematics Seminar, July 9» 1969
R. S. Kraft obtained a constructive existence proof for a pure initialvelue prol.lem associated witli a.linear transport equation. A paper cn tlJ.s subjecc i.s b^ing prepa,red. Ideas fci' deriving aprioriboimds for an approximation scheme associated with a bouirinrylniTlal problem for a linear transportequation have been obts.ined.
Tlie study of a finitedifference scheme for the KemstPlanck equations is continuing. Tecimlqucs foreDUablishing the positivity and convergence cf the nonlincar luerates are being examined.
R. S. Kraft discovered also a possibility for generalizing a method of reflecting solutions of elliptic
systems. The method, if feasible, would allow continuation across an analytic hcuncKu'y whon the solution of the system satisfies very general condi.tions on the boundary.
H. J. Oser completed, joi.ntly with J. E McKinney of the Rheology Section (21305), two manuscripts
deali.ng v,’ith acoustic propagation in viscous and inviscid, heatconducting media. The authors show
that certain solutions of the underlying fourthorder i)aitial differential equation are not stable, a
result which has not been I'epcrted heretofore in the ^Litci'abure. In viscous fluids the instability
does not occur if tlie vises; Ivy exceeds a certain critical, value wliich depends on the ratio of the
specific heats, the Prandtl number, and the viscosity.
12
Continuing his effort to provide au :lMproved autoicatic nincerlcnl intcgratio:i routine for OilinTAB,D. V;, Lozier produced end chocked out a Fortran program that uaca on adaptive Simijcon procedure. Healso completed and submitted to American University a paper entitled "Frechet Ill.fferential Calculus"as the finaJ. req^ulrement for the MA degree, which vas awarded in August. In another project, he developed an algorit)m, and produced an Iniplementlng Fortran program, to determine v/itli a iiigh degreeof precision the volume of a nearly perfect si^here from experimental data taken by measuring deviations from roundnecs of great circles on the sphere. This work was done for P. E. Pontius and H. A.Bovnoan of 213.31 (Mass and Volume Section, Metrology Pivlslon). Several test cases were run to checkout both the algorithm a;id the program. Finally, he provided jnatheraatical e,nd computer programmingsupport for W. L. Sadowski and other staff members of UBS.
Z. G. Ruthberg compai'ed the efficiencies of several matrix inversion programs and two programs forfinding roots of polynominals . This effort is intended to involve the Section mere deeply in theprogram for algorithm testing and development.
Z. G. Ruthherg also continued work on the binomi.ai operator [a(x,y) ^ + h(x,y) • A FORTRANprogram is now avaJ.lable that produces the successive powers n of this operator. Tliis new programis fai* more efficient (concerning the storage requirement) and faster than the earlier LISP program.
Consultations to and cooperative activities with other Bui*eau units continued throughout the period.These involved contacts with the Metrology ilLvision (now a part of the Atomic Physics Mvision), theMechanics Division, the Electronic Technology rivlsicn and the Meat Division. The Mass and VolumeSection of the Metrology Di.vision Js concerned with determining with high precision the volume ofalmost perfect spheres . A method vas developed with that group to determine the volume correctionsnumerlcal].y from a suitable number of traces along approximate great circles of the pseudo sphere.Tlie results appear to offer a nev; and considerably more economical way of obtaining precise volumes forpui'poses of density and el?.ss determinations.
Anothei’ effort for the Laser group in the Metrology U’.vision le devoted to checking whether anomalouslight propagation may take plr.ee in cerbaln laser optical arrangements . There is some speculationthat such might he the case. The investigations are not conclusive so far.
Work with J. E, McKinney ( 213. 05) was completed on the propagation of sound in viscous and inviscidmedia. A paper dealing with viscous case will appear in Jour, of Acoustical Society of America inthe January 1970 issue. . The paper on the inviscid case was subriiitted to the V/ashington EditorialReview Board for approval to appear in the KBS Journal of Research.
Shorter consultations with mejnbers of the Heat Division and the Electronic Technology Division involvedthe solution of Integral equations and boondary vad.ue problems
.
H. J. Oser seivcd as editor and trons]iitor of a book on the theoiy and computation of hoimdary layers,published by M.I.T. Press Inc.
Publications :
(1) Stability of Acoustic Waves witkin a Viscous, Coraprossible. HeatConductirjg Fluid. J. E. McKinneyand H. J. Oser. To appear in the Jour, of Acoustical Society of America, Vol. 47, Wo. 1. Jan.
1970.
(2) A. V/alz (edi.ted and transljated by H. J. Oser): Boundary Layers of Flow and Temperature. M.I.T.Press, Inc., October 19d9
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M/O’IffiliATICAL m'HOEG HI FUSKA PHYSICS
Task 2050412205014o/59M(2II5I1O
Origin: KBS /luthorized 10/].3/6lMejiager: VJalter L. SadovskiFull task description; OctoberEeceiater I96I issue, p. 12
A hlghordor Taylor series expenslon in the time vsxiable for the numerical solution of the Vlasovequation has been developed. The expansion pei'nlts the use of a smaller number of Hemlte poljTiomlnalsand has a very' sma^J. truncation error. The error is expllcltely' calculated. In addition to thehigher speed, the algorithm makes it possible to study truncation errors in both Fourier ar/d Hermltespace
.
The results show that the use of the electric field as 0 ci'iterion for numerical stability' is wholly'Inadequate, numerically' unstable runs giving substantially the same value of the electric field asstable runs. This Indicates that nujnerical studies of processes that are highly dependent on kineticeffects in plasma, such as hump on tall, must he reexamined to Insure that the results obtal.ned werenumerically accurate.
Work is in progress to determine the ridnlmum size of eigenfunction expansion sets for various values ofparameters. A paper on this topic is in preparation.
A oneday vrorkshop was held at I'lBS on October 15, 19^9 • The 'workshop dealt with the numerical solutionof the Vlasov equation. Amorg the topics discussed were the relative merits of the different methodsof solving the equation, reliability of results, numerical difficulties encountered and ways of dealingwith these.
Biring this conference the paidiclpaivts decided that a file of preprints should be held at HB3 and anewsletter be sent out to the participants to keep thera Informed on wliat the various groups aroimd the
country are doing. Ho agreement on Intercomporison of results has been reached, but a t’wov.ieek working
session, possibly at I'fflS, has been suggested for the summer of 1970 and the matter will be taken upagain.
Publications
(1) Temperature Renormalization in the NonLinear Vlasov Equation. W. L. Sadovski and Z. G. Ruthberg.
J. of Reseai’ch NBS, Vol. 73B, pp. 28I29I, Dec. I969.
(2) Selective E>:citatlon of Harmonics in a CoUisionless Plasma by Two Countorstreajnlng Electron Beams.W. L. Sadovski and Z. G. Ruthberg. J. of Reseta'ch NBS, Vol. 73B, pp. 293300, Dec. I969.
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Task 20500122050'i04
51560
Origin: NBSSponsor: National Institutes of HealthManager: Russell A. KirscliFull task description: JanuaryJune 196A issue. 19
Autliorized 1/21 /6A
Status: CORTpiUED, The general procedure developed on the Q32 computer for automatic segmentationof quantized images which yields as output a partial ordering (induced by the inclusion relation)of all morphological objects in an image was continued as elements of this partial ordering servedas inputs for a higher level syntactic analysis. These procedures were applied to optical serialsection photographs of a double neuron which v/ere made using a progr'r.mmablc microscope stage andfocus connected to the LINCFDP8 computer. The sections v;ere the’.i reconstructed and the morphologicanalyzer was also used to reconstruct an optical intensity photograph of two c’aromosomes
.
SCANNING MICROSCOPE PROCESSING
Task 205002050408
51560
Origin: NIiS Authorized 6/18/68Sponsor: National Institutes of HealthManager: Russell A. KirschFull task description: JulyDecember 1968 issue, p. 14
Status: CONIINITED. Progress was made in the study of the use of a programmable scanning microscopecontrolled by a PDP8 digital computer.
CHEMICAL BIOLOGICAL INFORMATION PROCESSING
Task 205002050410
51560
Origin: NBS Authorized 6/27/68Sponsor: National Institutes of HealthManager; Russell A. KirscliFull task description: JulyDecember' 1968 issue, p. 14
Status: CONTINUED. Further efforts v;ere made to coordinate the use of remote computers in communicating pharmacological information regarding chemical structure and biological activity among a groupof collaborating pharmacologists. Advisory work continued on Chemiccl Biological Information Handlintechniques
.
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Task PO5OOI1O205O] 2 .1 /57216
11550
Origin end Sponsor; NBSManager: I. A. StegunFull task description: July Pecern’oer 196)4 issue, p. )4
.
Stetus; COHTU'TUED. Consultation continued with the steff of KBS, other government agencies, Industryand universities. The topics covered were tables of specie! functions, general computing techniquesend compute tional pitfalls,
I. A. Stegun end R. Zucker, with emphasis on specie! functions, have conblnued(l ) the file maintenanceof available algorithms, spproximetlons and programs (2) the critical exemlnetlon of availablealgorithms end opiproxlraations (3) the testing of the implementing subroutines. A paper has beenprepared on automatic computing methods for special functions in particular the error and complementary error function, The FORTRAN program yielding full machine accuracy for both functionsover the entire range will be presented together \;ith the test program and sample results.
In the course of the testing of subroutines, hardware and software errors have lieen detected, inparticular, double precision mul tiplicetJ.on yielded incorrect single precision results under certainconditions; the compiler set up the computation of an external factor as an internal factor in'a suriimation. In collaboration with the Center foi' Computer Sciences and Technology, the hardwareerror was corrected end the software error is being further analysed.
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Oav/son, J.
Armstrong, T,
Feix, Iih
Danov i 1 1 , J
.
Mere I', H.
Kruskal, M.
Symoii, K.
Sadov,ski, W„
Sliero, Kenneth D.
Gilsinu, David
Marlin, Monroe H.
ilOGBEN, D,
WALKER, J. C.
OLVER, F, W. ,J.
SADO’.VSKI, W. L.
JOEI., L. S.
Lectures jchnical Mr *' rra
Workshop on Vlasov Equations
October 15, 19G9
discussed were as follows
:
(Princeton University) Pi'.rticle v,s. distribution function inodojs
(University of Kansas) Interconiparison of eigenfunction oxjiansion with theLos Alamos codes
(University of Nancy, France) Tlio waterbag inodel
(NRL) FourierFourier transform niothod for the solution of the Vlasov equation
(Oak Kidge i^aboratory) On some aspects of truncation in FourierFouriortransforms of the Vlasov equation
(Princeton University) Some remarks on the f;tability
(Madison, Wisconsin) Bit pushing for simulation of particle models
(Systems Dynamics Section) Setting up of a coordination center and the problc;.of intercomparison of numerical lesults
Systems Dynamics Seminal'
NASAGoddard Space Flight Center and University of Nebraska, Lincoln, Nebraska.Oscillation Theorems for a Class of Delay Equations. July 23.
Melpar, Inc. (A Division of .American Standaid Co,), Falls Church, Va . Domainsof Stability of Integral Manifolds for Some Nonlinear Differential Equations.•Nov. 12,
Institute for Fluid Dynamics, University of !\!arylar.d. A Now Method for FindingExact Solutions of the NavicrStokes Equations. Doc. tO.
Mathomatics Division Expository L.c t nj_e_s
Computing for the Countless. July 23.
(Center for Computer Sciences and Teclmology) Categorical Algebra as a New
Approach to Mathematics. September 17.
Tlie Method of Chester, Friedman and Urseli Asymptotic E.ypansion oi Definite
Integi'als Having Two Neai’ly Coincident Sand lo Poi n ts . October 22.
The Numerical Solution of Differential Equations uy an NthOixler Ku.nge Ju t ta
Method. Deceinber 3.
Loose Ends in tlie Tlieory of Games. Deterabei' IV.
NBS Graduate School,Fall .Semester
PEAtCf, S. T. Fortran Programming for Beginners
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HOGBEN, David Computing for Die countless. American Statistical Association, New Y'ork,August 19.
JOINER, Brian L. The use of OMNITAB foi statistical analysis of tabular data. Amei'icauStatistical Association, Conierence on Small Area Statistics, New Yoik,August 21,
KIRSCM, R. A. Using Computers as Visual rather than verbal devices. D, C. Teachers'College, October 16.
KIRSCII, R. A. Computer determination of the constituent structure of biological images.Computer Science colloquium, Rutger.s University, December 5.
KRAFT, R. S. Convergence of semidiscrete approximations of linear tr.ansport equations.University of Oregon, Eugene, Oregon, August 29,
KU, H. H Analysis of multidimensional contingency tables: An inforruation theoreticapproach. International Statistical Institute, London, September 5.
NEWMAN, M. Number theoretic subroutines lor high speed digital com])uters, O.xfoid,England, August 20.
Computational problems connected with certain matrix groups, Oxford,England, August 21.
Normal subgroups of the modular group, Madison, Wisconsin, October 28.
VARNER, Ruth N. OMNITAB. Washington Statistical Society and Urban Regional InformationSystems Association, Septcml)er 30.
18
PUBLICATIONS ACTF/ITIUS
1.0 PUOLICATIONS .THAT APPEARED DURING THIS PERIOD
1.2 Monograph
Lr;ctures on Modular Forms. J. Lehner, AMS 61.
l.H Technical Papers
Minimax Error Selection of a Univariate Distribution witli Prescribed Componentwise Hounds andHanking. A. J. Goldman. J. of Research KBS, 73B 0939), pp. 2252.:l0.
Minimax Adjustnicnt of a Univariate Distribution to Satisfy Cornpohentwiso Bounds and/or Ranking.A. J. Goldman. J. of Research NES, 73B (19G9), pp, 231239.
Simultaneous Contractification. A. J. Gol. L. Sadowski and 2, G. HutlibcrJ. of Research NBS, Vol. 7^. No. 4, pp. 281291 , (Dec . 1969).
Selective Excitation of Harmonics in a Col 1 is i onless Plasma by Two Counterstrcariing Electron
Beams. W. L. Sadowski and Z. G. Uutliberg. J. of Research NBS. Vol. 73D , No. 4, pp. 293300,
(Dec. 1969).
An Evaluation of Linear Least Squares Computer Programs. Roy H. Wampler, J. of ResearcliNES, 7^ (1969) 5990.
19
1 » 5 Books
A. Walz (oditocl and IransJatod by li. J. Osor) : BovnuUiry Bayers o.t Flow and Temperature.M.l.T. Press, Inc. Octolier ]969.
2.0 mNuseniPTs in tup process of publication
' 2.3 Technical Papers
INormal Subgroups of the Modular Group, I.. Greenberg and M. Nev/man, Submitted to a leelmicnj
I journal,
i;
Some Results on Solvable Groups. L, Grcenbe)g and M, Newman. To ajjpear in Arcli. der MaUi./
Sequences of Numbers that are Approximately Completely Equi dis tribu t cd . S, Haber. To ai)neaiin J. of Assoc, for Computing Machinery.
I
Niunerical Evaluation of Multiple Integrals. S. Haljer. Submitted to a tcclinieal joujnal .s y t' On the Sum L and Numerical Integration. S. Haber and C. F. Osgood. To appea: in
Pacific J. of Math.
Symn,ctry and luarginal Homogeneity of an r x r Contir.gency Talilo, C. T. Ireland, S. Kuilba
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A Report on the Accuracy of Some Widely Used I
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PERIODICALS
JOURNA.L OF RESEARCH reports NationalBureau of Standards research and development inphysics, matliematics, chemistry, and engineering.Compreliensive scientific papers gi\’e conijjlete. detailsof tiie work, including laboratory data, ocperimcntalprocedures, and theoretical and mathematical analyses. Illustrated with photographs, drawings, andcharts.
Published in three sections, available separately:
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Pajrers of interest primarily to scientists working in
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physical and chemical research, with major emphasison standards of physical measurement, fundamentalconstants, and properties of matter. Issued six timesa year. Annual sub.scription : Domestic, $9.50; foreign, $11.75*.
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Studies and compilations designed mainly for themathematician and theoretical physicist. Topics inmathematical statistics, theoiy of experiment design,
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TECHNICAL NE¥/S BULLETIN '
The best single source of inform. at ion concerning theBureau’s research, developmental, cooperati\e and
publication activities, this monthly publication is
designed for the industiyoriented individual whose
daily work involves intimate contact with science and
technology
—
for caigincrrs, chemists, physicists, re
search managers, productdevelopment managers, and
company executives. Annual subscription: Domestic,
$3.00; foreign, $4.00*.
DifTcronce in price is due to extra coat of torciun roailinp.
Order NBS publications from:
Applivcd Mathematics Series. Mathematical tables,manuals, and studies.
Building Science Series. Research results, testmethods, and performance criteria o! building nia
terials, components, systenrs, and structures.
Handbooks. Recommended codes of engineeringand industrial practice (including safety codes) developed in cooperation with interested industries,
professional organizations, and regulatory bodies.
Special Publications, Proceedings of NBS conferences, bibliographies, annual reports, wall charts,
pamphlets, etc.
Rfonographs. Major contributions to the technicalliterature on various subjects related to the Bureau’s
scientific and technical activities.
National Standard RefereEce Data Series.NSRDS provides cjuantitafive data on the physicaland chemical propertivcs of materials, compiled fromthe world’s literature and critically evaluated.
Product Standards, Provide requirements for sizes,types, quality and methods for testing various indus
trial products. These standards are developed coopera
tively with interested Government and industry groupsand provide the basis for common understanding ofproduct characteristics for both buyers and sellers.Their use is voluntary.
Technical Notes. Thi': series consists of communications and reports (covering botli other agency andNBSsponsored work) of limited or transitory interest.
Federal Irsformation Processing Standards PublicatiorES, This series is the official publication within
the Federal Government for information on standardsadopted and promulgated under the Public Law89—306, and Bureau of the Budget Circular A—86entitled. Standardization of Data Elements and Codesin Data Systems.
Superintendent of DocumentsGovernment Printing OfficeWasliington, D.C. 20402
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